Trifoliolate

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Trifoliolate is a single-axis, single-prototile hexagonal deformation designed by Glen Paris at the studio of William Huff in 1966. Dataflow diagram of the prototile is defined by first implementing a “manual” Euclidean construction of ruler and compass, then this applied to Grasshopper using a curve evaluation method (which is much optimal). This dependency graph reveals parametric potentials of the tiling. After that, a gradial manipulation is added to the tiling in order to create the original parquet deformation. Finally, further deformation opportunities are experimented on the tiling. Dataflow modeling is based on the hexagonal hyperframe underneath.

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Here is a small phrase from Hofstadter, talking about this pattern deformation:

Three of my favorites are “Crazy Cogs” (Figure 10-7, done by Arne Larson, Carnegie-Mellon, 1963), “Trifoliolate” (Figure 10-8, done by Glen Paris, Carnegie-Mellon, 1966),and “Arabesque” (Figure 10-9, done by Joel Napach, SUNY at Buffalo, 1979). They all share the feature of getting more and more intricate as you move rightward. Most of the earlier ones we’ve seen don’t have this extreme quality of irreversibility-that is, theratcheted quality that signals that an evolutionary process is taking place. I can’t helpwondering if the designers didn’t feel that they’d painted themselves into a corner,especially in the case of “Arabesque”. Is there any way you can back out of that super-tangle except by retrograde motion-that is, retracing your steps? I suspect there is, but I wouldn’t care to try to discover it.

Hofstadter, D., 1983, “Metamagical Themas Questing for the Essence of Mind and Pattern”, Chapter 10. (can be read here)

When you switch the x and y axes of the plane, strange patterns start to emerge:

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